Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph

We prove that after an arbitrarily small adjustment of edge lengths, the spectrum of a compact quantum graph with δ-type vertex conditions can be simple. We also show that the eigenfunctions, with the exception of those living entirely on a looping edge, can be made to be non-vanishing on all vertices of the graph. As an application of the above result, we establish that the secular manifold (also called “determinant manifold”) of a large family of graphs has exactly two smooth connected components.